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Lecture 2-1
COULOMB FORCE
q1q2
F1,2 k 2 rˆ1,2
r1,2
“Test Charge” q2
Charge q1
Question: How can q1 sense q2
without direct contact?
r
“Is action at a
distance possible?”
• NO! just as air around us, there’re
some medium around a charge, the
electric field that transfers the force.
Lecture 2-2
READING QUIZ 1
HOW DOES THE MAGNITUDE OF THE ELECTRIC FIELD E
FROM A POINT CHARGE Q DEPEND ON THE MAGNITUDE OF Q
AND THE DISTANCE R TO THE FIELD POINT?
A) E = k Q Q / R2
A) E = k Q / R
B) E = k Q / R2
C) E = k Q Q / R
A) E = k Q / R3
Lecture 2-3
Electric Field
Define electric field, which is independent of the test charge,
q2 , and depends only on position in space:
F
E
q
Electric Field due to a
Point Charge Q
F
1 Q
E 
rˆ
2
q 4 0 r
Lecture 2-4
Electric Field Lines
• A visualization tool to illustrate the geometry of
an electric field.
• Electric field lines originate from positive charges
and terminates at negative charges. (or at infinity)
• The direction of the electric field at any location is
tangential to the field line there.
• The magnitude of the electric field at any location
is proportional to the density of the lines there.
# lines/area ~ 1/r2 ~ |E|
Lecture 2-5
Electric Field
With this concept, we can “map” the electric field anywhere in space
A Vector field
73
77
72
71
82
84
83
88
75
68 64
80
73
57 56 55
66
88
75 80
90
83
92
91
77
Lecture 2-6
Demonstration
•
Vandegraaff Generator
Lecture 2-7
Electric field lines of two charges of equal magnitude
dipole
Far from the charges, electric
field lines resemble those of a
point charge.
 Field lines do NOT cross
 They do NOT come out of
nowhere or vanish into a point.
Lecture 2-8
Two charges of the same magnitude
• Both charges > 0
• One is > 0, the other < 0
-q
d
E
q
electric dipole of
dipole moment:
p  qd
if |qR|>|qL|
2kp
x3
if |qL|>|qR|
Lecture 2-9
Electric field lines of two unequal charges
Far from charges, the
field lines are as if
they are due to a point
charge of +2q-q=+q
# lines proportional to the
magnitude of charge
http://www.its.caltech.edu/~phys1/java/phys1/EField/EField.html
http://www.falstad.com/vector3de/
Lecture 2-10
©2008 by W.H. Freeman and Company
Lecture 2-11
Electric Field due to Multiple Point Charges
The force on a test charge is then given by
  

F  F1  F2  F3  
so the electric field is, by definition, given by




 F F1 F2 F3
E     
q0 q0 q0 q0



 E1  E2  E3  
Principle of Superposition!
Lecture 2-12
Lecture 2-13
Lecture 2-14
( Sum symbol Σ )
Σ Fx = 3.97 x 10-7 N
E = F/qO
Σ Fy = - 2.77 x 10-7 N
F2 = Fx2 + Fy2 = (4.84 x 10-7 N)2
Tan Θ = Fy /Fx = - 34.9 O
Lecture 2-15
Warm-up quiz 2
What is the direction of the electric field at the center
of the two point charges? (Take Q > 0.)
a)
b)
c)
d)
e)
toward the lower right
toward the upper left
upward
to the right
none of the above
Q
-Q
Lecture 2-16
Point Charge in an external Electric Field
• Electrostatic force on the charge q due to E: F  qE
F is parallel to E if the charge is positive and
anti-parallel if the charge is negative
• E acting on the charge is produced by the other charges
(external field).
The charge is not affected by its own electric field.
• Dimension of E = [force]/[charge] and
its SI units
= N/C
• Work done by E on q in moving the charge:
WAB  
: A B
qE dl
W is path( )-independent
=> Conservative force
Lecture 2-17
Demonstration
• Dynamics of a charged Mass in an Electric field
Lecture 2-18
Dynamics of a Charged Mass in Electric Field
For -Q<0 in uniform E downward:
F  ma  ( Q ) E
QE
a  ay j 
j (E  E j)
m
1 2
 y (t )  a y t , x (t )  v x t
2
-Q
2
1  x  QEx 2
y  ay   
2
vy2 = at
=
qE/m
t
vx >>0
v
2
mv
x
 x
 QEt 
v (t )  v  v y (t )  v   
 m 
tan   y
2
xv
2
2
x
2
• Oscilloscope
• Ink-Jet Printing
• Oil drop experiment
vx
http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/electric_single_particle/applet.html
Lecture 2-19
The Electric Dipole
Along the + z-axis
Where p=qd is the magnitude of
the dipole moment. The electric
dipole is a vector in the direction
from –q to +q. Let d 0 keep qd
constant.This a point dipole
Lecture 2-20
Dipole in a uniform external electric field
• No net force. The electrostatic
forces on the constituent point
charges are of the same
magnitude but along opposite
directions. So, there is no net
force on the dipole and thus its
center of mass should not
accelerate.
Clockwise about
the center of mass
• Net torque! There is clearly a net
torque acting on the dipole with
respect to its center of mass, since
the forces are not aligned.
http://qbx6.ltu.edu/s_schneider/physlets/main/dipole_torque.shtml
Lecture 2-21
Torque on the Dipole
F
The magnitude of the torque:
  Fx sin   F (d  x) sin 
 Fd sin   qEd sin 
 pE sin 
The torque points into the screen.
 
  p E

x
CM
F

Lecture 2-22
Electric Dipoles
• Typical dipole consists of positive and
negative charges slightly displaced.
• General definition of dipole moments
exists:
p   ( r )rd 3r
 

• Water molecule can be thought of
as consisting of 2 standard dipoles at
an angle to each other.
 Net neutral molecules can have
electrical dipole moments
 Permanent dipole moment (polar)
vs. induced dipole moment
Lecture 2-23
Physics 241 – 10:30 Quiz 3 – January 14, 2010
Four point charges are arranged at the corners of a
square as shown. What is the direction of the electric
field at the center of the square? (Take Q > 0.)
a)
b)
c)
d)
e)
toward the lower right
toward the upper left
upward
to the right
none of the above
-Q
-Q
Q
Q
Lecture 2-24
Physics 241 – 11:30 Quiz 3 – January 14, 2010
Four point charges are arranged at the corners of a
square as shown. What is the direction of the electric
field at the center of the square? (Take Q > 0.)
a)
b)
c)
d)
e)
toward the lower right
toward the upper left
upward
to the right
none of the above
-Q
Q
Q
Q
Lecture 2-25
Physics 241 – Sample Quiz C – January 9, 2008
Three point charges are arranged at three of the corners
of a square as shown. What is the direction of the
electric field at the center of the square? (Take Q > 0.)
a)
b)
c)
d)
e)
toward the lower right
toward the upper left
upward
to the right
none of the above
Q
Q
Q